### All GMAT Math Resources

## Example Questions

### Example Question #1 : Dsq: Calculating The Length Of A Radius

Give the radius of a circle on the coordinate plane.

Statement 1: A square whose vertices include and can be inscribed inside the circle.

Statement 2: A right triangle whose vertices include and can be inscribed inside the circle.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. The length of a segment with the given endpoints can be calculated using the distance formula. However, it is not clear whether the points are opposite vertices, in which case the segment is a diagonal of the square, or the points are consecutive vertices, in which case the segment is a side of the square, making the diagonal of the square times this length. The length of the diagonal of the inscribed square cannot be determined for certain; since the diameter of the circle is equal to the length of the diagonal, the diameter cannot be determined, and since the radius is half this, the radius cannot be determined.

Assume Statement 2 alone. The length of a segment with the endpoints can be calculated using the distance formula. However, it is not clear whether the segment is a hypotenuse of the triangle or not; the diameter of a circle is equal to the length of the hypotenuse of an inscribed right triangle, so knowing this is necessary.

Assume both statements to be true. The two statements together give four points of the circle; since three points uniquely define a circle, the circle can be located; subsequently, the radius can be found.

### Example Question #2 : Dsq: Calculating The Length Of A Radius

Rectangle is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

**Possible Answers:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The diameter of a circle that circumscribes a rectangle is equal to the length of a diagonal of the rectangle; the radius is equal to half this.

The two statements together, however, do not yield this. The opposite sides of a rectangle are congruent, so the two statements are actually equivalent; each gives the same dimension of the rectangle. This is insufficient to determine the length of the diagonal.

### Example Question #3 : Dsq: Calculating The Length Of A Radius

An equilateral triangle is inscribed inside a circle; is the midpoint of . What is the radius of the circle?

Statement 1: has area .

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

First, locate the other midpoints of the sides of the triangle and construct the segments from each vertex to the opposite midpoint.

Since is equilateral, , , and are all altitudes that insersect at the center of the circumscribed circle, , so that . is the radius of the circumscribed circle.

Assume Statement 1 alone. The length of one side of an equilateral triangle can be calculated using the formula

, or, equivalently,

Once is calculated, then, since is also a perpendicular bisector of and a bisector of , making a 30-60-90 triangle, can be calculated to be one half of ; can be multiplied by to yield , and, since the three altitudes of an equilateral triangle divide one another into segments whose lengths have ratio 2:1, can be multiplied by to obtain radius .

Statement 2 gives us explicitly, so we can take two thirds of this to get the radius

.

### Example Question #4 : Dsq: Calculating The Length Of A Radius

Between Circle 1 and Circle 2, which has the greater radius?

Statement 1: A arc of Circle 1 has length equal to one fourth the circumference of Circle 2.

Statement 2: A sector of Circle 2 has area equal to four ninths that of Circle 1.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. A arc of Circle 1 has the circumference of Circle 1. Since this is also the circumference of Circle 2, then, if we let be the circumferences,

,

and

.

This gives Circle 2 the greater circumference and, subsequently, the greater radius.

Assume Statement 2 alone. A sector of Circle 2 has area of Circle 2. Since its area is also equal to that of Circle 1, then, if are the areas of Circle 1 and Circle 2, respectively, then

,

and

This gives Circle 2 the greater area and, subsequently, the greater radius.

### Example Question #5 : Dsq: Calculating The Length Of A Radius

Give the radius of a circle on the coordinate plane.

Statement 1: A right triangle with a hypotenuse with endpoints and can be inscribed in the circle.

Statement 2: A right triangle with a leg with endpoints and can be inscribed in the circle.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

If a right triangle can be inscribed inside a given circle, then its hypotenuse has a length equal to the diameter of the circle, and the radius of the circle can be calculated as half this. Statement 1 gives sufficient information to find this, since the length of the hypotenuse is the distance between its endpoints and , which is ; the diameter of the circle is 20, and the radius is half this, or 10. From Statement 2, we can only find the length of one leg of an inscribed right triangle, so the length of the hypotenuse is still open to question.

### Example Question #441 : Geometry

Parallelogram is inscribed inside a circle. What is the radius of the circle?

Statement 1: Each side of Parallelogram has length 20.

Statement 2: .

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

Opposite angles of a parallelogram are congruent, and if the parallelogram is inscribed, both angles are inscribed as well. Congruent inscribed angles on the same circle intercept congruent arcs; since the two congruent arcs together comprise a circle, each intercepted arc is a semicricle. This makes the angles right angles, and this forces a parallelogram inscribed in a circle to be a rectangle.

Statement 1 alone tells us that this is also a square, and that its sides have length 20. The diagonal of a square, which is also a diameter of the circle that circumscribes it, has length times that of a side, or ; half this, or , is the radius of the circle.

Statement 2 alone gives a diagonal of the rectangle, which, again, is enough to determine the radius of the circle.

### Example Question #12 : Circles

A circle is inscribed inside an equilateral triangle . , , and are tangent to the circle at the points , , and , respectively. What is the radius of the circle?

Statement 1: The length of arc is .

Statement 2: The degree measure of arc is .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The figure referenced is below:

By symmetry, is one third of a circle. Therefore, its length is one third of the circumference, so, if Statement 1 alone is assumed, the circumference can be determined to be ; this can be divided by to yield radius .

Statement 2 yields no helpful information; from the body of the problem, can already be deduced to be two thirds of a circle, or, equivalently, an arc of measure

.

### Example Question #11 : Circles

Square is inscribed inside a circle. What is the radius of the circle?

Statement 1: Square has area 100.

Statement 2: .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

From Statement 2 alone, , a diagonal of the square, measures . The diameter of the circle is equal to the length of a diagonal of an inscribed square, so the radius of the circle is equal to half this, or .

From Statement 1 alone, since the area of the square is 100, its sidelength is the square root of this, or 10. By the 45-45-90 Theorem, a diagonal of the square measures times this, or , which makes Statement 2 a consequence of Statement 1. Therefore, it follows again that the circle has radius .

### Example Question #11 : Circles

Right triangle is inscribed inside a circle. What is the radius of the circle?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

If a right triangle is inscribed inside a circle, the hypotenuse of the triangle is a diameter of the circle; therefore, its length is the diameter, and half this is the radius.

Statement 1 alone does not give the hypotenuse of the triangle, or, for that matter, any of the sidelengths. Statement 2 alone gives one sidelength, but does not state whether it is the hypotenuse or not.

Assume both statements are true. Since in right triangle , then either and , or vice versa. In either event, , being opposite the angle, is the short leg of a 30-60-90 triangle, and, by the 30-60-90 Theorem, the hypotenuse is twice its length. This is twice 18, or 36. This is the diameter of the circle, and the radius is half this, or 18.

### Example Question #15 : Circles

Give the radius of a circle on the coordinate plane.

Statement 1: The circle has its center at .

Statement 2: The circle has its -intercepts at and .

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Knowing neither the center alone, as given in Statement 1, nor two points alone, as given in Statement 2, is sufficient to find the radius of the circle.

Assume both statements to be true. Knowing the center from Statement 1 and one point on the circle, as given in Statement 2, is enough to determine the radius - use the distance formula to find the distance between the two points and, equivalently, the radius.

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